I am searching for a book of this form and content, is there any?

Question

I would like to know is there a book that is both a history of mathematics and a collection of open problems?

I know that there exist many books that cover either larger or smaller periods of the history of mathematics and also a number of books that are collections of open problems, but, a symbiotic approach I have not seen.

So, a book that I would like to read would inform us about development of mathematics at least from a period of Greek mathematics to the twentieth century, and it would be a story also about some specific problems that led to the development of open problems that are not resolved till the present day, and that book could end with Hilbert problems.

I think that development of open problems is also an important aspect of the history of mathematics, and such a book could be filled with some "high" mathematics, but not with too much of it, so that it does not diverge from the historic and informational character and it does not turn into a blending of "high" mathematics with some historical remarks and content.

So, I prefer that it does cover open problems from "higher" mathematics but written from an elementary standpoint, so that it more involves a description of open problems than their advanced formulation.

So, is there some book that closely resembles this description?

Answer 1

I think you would be hard pressed to find one book that covers this whole period for all topics. For example, The Code Book (by Simon Singh) is a wonderfully written book which covers the history of the subject and ends with the the current problems in the topic which are being researched (i.e. quantum computing).

If it's more the open problems part you are looking for, I would suggest looking at the Millennium Prize Problems and researching the history behind them.

Answer 2

I can recommend Colin C. Adams: "the knot book" for a very accessible introduction to knot theory. It covers some history of knot theory, esp. of knot tabulation, in the first few chapters, mentions many open problems that are at least easy to understand (I could even solve one of them, and I was a beginner), and contains many exercises.

Like @Bee, I think it will be difficult to find a broader book with a similar taste. Well, Dominic Olivastro's "ancient puzzles" is very broad and presents one (1!) open problem in graph theory at the end.

In theoretical computer science, "the art of computer programming" by Donald E. Knuth follows the approach you describe, developing each of the treated problems from their historic perspective. But many people (including me) find it hard to read, although both the writing and the coverage of the whole history of ideas is excellent. Well, I admit that I skipped most of the literally thousands of exercises, which are presented along with accessible research problems. If you dare, I'd recommend looking first into volume 4A, which contains more combinatorial puzzles and less low-level bit-fiddling than the other volumes.